Quantum affine algebras and holonomic difference equations

Abstract

We derive new holonomic q-difference equations for the matrix coefficients of the products of intertwining operators for quantum affine algebra[Figure not available: see fulltext.] representations of level k. We study the connection opertors between the solutions with different asymptotics and show that they are given by products of elliptic theta functions. We prove that the connection operators automatically provide elliptic solutions of Yang-Baxter equations in the "face" formulation for any type of Lie algebra {Mathematical expression} and arbitrary finite-dimensional representations of[Figure not available: see fulltext.]. We conjecture that these solutions of the Yang-Baxter equations cover all elliptic solutions known in the contexts of IRF models of statistical mechanics. We also conjecture that in a special limit when q→1 these solutions degenerate again into[Figure not available: see fulltext.] solutions with {Mathematical expression}. We also study the simples examples of solutions of our holonomic difference equations associated to {Mathematical expression} and find their expressions in terms of basic (or q-)-hypergeometric series. In the special case of spin -1/2 representations, we demonstrate that the connection matrix yields a famous Baxter solution of the Yang-Baxter equation corresponding to the solid-on-solid model of statistical mechanics. © 1992 Springer-Verlag.